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The following is immediate from Theorem 3.5.3 and 4.3.4:

Corollary 4.3.5. There is a sequential asynchronous process coalgebra of type I, O that is ﬁnal in SAPC I,O.

4.4 Axiomatization Corollary 4.3.5 is the basis for our axiomatization. Essentially, what are axioms state are that processes are members of a sequential asynchronous process coalgebra of type I, O that is ﬁnal in SAPC I,O. This approach to the deﬁnition of a process is inspired by [2] and [57].

We will not be too obsessed with formalistic issues, as we want these axioms expressed in the same manner they are to be used.

In the following, we will used accented and subscripted variants of p for processes. We will write P, for the class of all process and the cooperation of the corresponding coalgebra.

Essentially, this is a metalinguistic device to refer to the model of the theory from inside the theory. This will enable us to express an extremal axiom that guarantees that are coalgebra is ﬁnal in SAPC I,O. And the consistency of that axiom will be a consequence of Corollary 4.3.5.

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Axiom of Unimpeded Input. For every p and any i, m ∈ actin I, O, there is p such i,m that p −→ p.

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The Extremal Axiom can further be reduced in the obvious way within the language of the theory using Theorem 3.5.3.

4.5 The Postulate of Delegated Output The Postulate of Delegated Output stands out, as it is the only one we have titled a “postulate” instead of an “axiom”. This is done with the purpose of initiating a discussion about its use and connection to the notion of asynchrony. Intuitively, even though it is practically assumed in every formalization of asynchrony that we know of, we would like to exclude it. Because we would like to consider computational processes running

CHAPTER 4. SEQUENTIAL ASYNCHRONOUS PROCESSES 106

asynchronously on a non-distributed system, and exchanging messages over say queues or buﬀer as asynchronous. But this creates problems.

In [14], Brock and Ackerman came up with their famous anomaly that proved that relational semantics was never going to be suﬃcient for the denotational characterization of interactive systems. The following example is from [54], and demonstrates two programs that have the same history relation, yet diﬀerent behaviour when in a feedback

conﬁguration:

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In [29], Jonsson proved that in order to obtain a compositional semantics, one must at least add enough information to describe the behaviour of a process as observed by a linear, sequential observer. And as it turns out, his result rests on the Postulate of Delegated Output.

The next example shows two diﬀerent components that display the same behaviour with respect to such an observer, but still behave diﬀerently in a feedback conﬁguration. The pseudocode below is understood as the program of a sequential process that reads from a buﬀer i and writes to a buﬀer o. Notice that reads are internal actions, while writes are external. And this is what violates the Postulate of Delegated Output: each of these two processes, when in feedback, after writing something to its own buﬀer, that written value is there to be read the next time a read is attempted. The reader is invited to verify that the traces of the two processes agree as they are, but disagree when in feedback.

CHAPTER 4. SEQUENTIAL ASYNCHRONOUS PROCESSES 107

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It is not hard to see that the problem is created only in direct feedback, and so perhaps it is instead that mechanism that is ill conceived. But more work is required to understand this new kind of anomaly.

Bibliography [1] Karl Abrahamson. Decidability and Expressiveness of Logics of Programs. PhD thesis, University of Washington at Seattle, 1980.

[40] Robin Milner. A calculus of communicating systems. Report ECS-LFCS-86-7, Laboratory for Foundations of Computer Science, Department of Computer Science, University of Edinburgh, August 1986. (First published by Springer-Verlag as Vol.92 of Lecture Notes in Computer Science).

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